Olympiad problems

2012 Serbia Team Selection Test, 2

Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.

2008 IMS, 8

Tags: modular arithmetic , number theory proposed , number theory

Find all natural numbers such that \[ n\sigma(n)\equiv 2\pmod {\phi( n)}\]

2007 JBMO Shortlist, 5

Tags: quadratics , modular arithmetic , number theory , number theory proposed

Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

2009 Bosnia Herzegovina Team Selection Test, 2

Tags: number theory proposed , number theory

Find all pairs $\left(a,b\right)$ of posive integers such that $\frac{a^{2}\left(b-a\right)}{b+a}$ is square of prime.

2013 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: number theory , greatest common divisor , modular arithmetic , algorithm , number theory proposed

Find all positive integers $n$ for which there exist two distinct numbers of $n$ digits, $\overline{a_1a_2\ldots a_n}$ and $\overline{b_1b_2\ldots b_n}$, such that the number of $2n$ digits $\overline{a_1a_2\ldots a_nb_1b_2\ldots b_n}$ is divisible by $\overline{b_1b_2\ldots b_na_1a_2\ldots a_n}$.

2005 CentroAmerican, 1

Tags: number theory proposed , number theory

Among the positive integers that can be expressed as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order? [i]Roland Hablutzel, Venezuela[/i] [hide="Remark"]The original question was: Among the positive integers that can be expressed as the sum of 2004 consecutive integers, and also as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?[/hide]

2001 Romania Team Selection Test, 1

Tags: modular arithmetic , number theory proposed , number theory , polynomial congruence

Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.

2010 Bosnia Herzegovina Team Selection Test, 1

Tags: number theory , prime numbers , number theory proposed

$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$. $b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.

2024 Middle European Mathematical Olympiad, 7

Tags: number theory , number theory proposed , Digits

Define [i]glueing[/i] of positive integers as writing their base ten representations one after another and interpreting the result as the base ten representation of a single positive integer. Find all positive integers $k$ for which there exists an integer $N_k$ with the following property: for all $n \ge N_k$, we can glue the numbers $1,2,\dots,n$ in some order so that the result is a number divisible by $k$. [i]Remark[/i]. The base ten representation of a positive integer never starts with zero. [i]Example[/i]. Glueing $15, 14, 7$ in this order makes $15147$.

2003 Kazakhstan National Olympiad, 1

Tags: number theory proposed , number theory

Find all natural numbers $ n$,such that there exist $ x_1,x_2,\dots,x_{n\plus{}1}\in\mathbb{N}$,such that $ \frac{1}{x_1^2}\plus{}\frac{1}{x_2^2}\plus{}\dots\plus{}\frac{1}{x_n^2}\equal{}\frac{n\plus{}1}{x_{n\plus{}1}^2}$.

1997 Baltic Way, 5

Tags: number theory proposed , number theory

In a sequence $u_0,u_1,\ldots $ of positive integers, $u_0$ is arbitrary, and for any non-negative integer $n$, \[ u_{n+1}=\begin{cases}\frac{1}{2}u_n & \text{for even }u_n \\ a+u_n & \text{for odd }u_n \end{cases} \] where $a$ is a fixed odd positive integer. Prove that the sequence is periodic from a certain step.

1978 IMO Longlists, 39

Tags: number theory proposed , number theory

$A$ is a $2m$-digit positive integer each of whose digits is $1$. $B$ is an $m$-digit positive integer each of whose digits is $4$. Prove that $A+B +1$ is a perfect square.

2006 Croatia Team Selection Test, 1

Tags: number theory proposed , number theory

Find all natural numbers that can be expressed in a unique way as a sum of ﬁve or less perfect squares.

2010 Abels Math Contest (Norwegian MO) Final, 4b

Tags: calculus , integration , number theory proposed , number theory

Let $n > 2$ be an integer. Show that it is possible to choose $n$ points in the plane, not all of them lying on the same line, such that the distance between any pair of points is an integer (that is, $\sqrt{(x_1 -x_2)^2 +(y_1 -y_2)^2}$ is an integer for all pairs $(x_1, y_1)$ and $(x_2, y_2)$ of points).

2007 Pre-Preparation Course Examination, 1

Tags: inequalities , limit , floor function , absolute value , number theory proposed , number theory

Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.

1993 All-Russian Olympiad, 1

Tags: number theory proposed , number theory

For a positive integer $n$, numbers $2n+1$ and $3n+1$ are both perfect squares. Is it possible for $5n+3$ to be prime?

2014 Contests, 3

Tags: number theory proposed , number theory

Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be [i]good[/i] if [list] [*] [b](i)[/b] $0\le a_i\le p-1$ for all $i$, and [*] [b](ii)[/b] $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and [*] [b](iii)[/b] $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.[/list] Determine the number of good $p$-tuples.

2014 Moldova Team Selection Test, 4

Tags: AMC , AIME , number theory proposed , number theory

Define $p(n)$ to be th product of all non-zero digits of $n$. For instance $p(5)=5$, $p(27)=14$, $p(101)=1$ and so on. Find the greatest prime divisor of the following expression: \[p(1)+p(2)+p(3)+...+p(999).\]

1993 Baltic Way, 5

Tags: number theory proposed , number theory

Prove that for any odd positive integer $n$, $n^{12}-n^8-n^4+1$ is divisible by $2^9$.

2001 ITAMO, 3

Tags: number theory proposed , number theory

Consider the equation \[ x^{2001}=y^x .\] [list] [*] Find all pairs $(x,y)$ of solutions where $x$ is a prime number and $y$ is a positive integer. [*] Find all pairs $(x,y)$ of solutions where $x$ and $y$ are positive integers.[/list] (Remember that $2001=3 \cdot 23 \cdot 29$.)

2010 Moldova Team Selection Test, 1

Tags: modular arithmetic , number theory proposed , number theory

Find all $ 3$-digit numbers such that placing to the right side of the number its successor we get a $ 6$-digit number which is a perfect square.

2014 China Team Selection Test, 6

Tags: induction , number theory proposed , number theory

For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$. Prove: If $n$ is a positive integer greater than $1$, $p$ is a prime factor of $n$, then $f(n)\leq \frac{n}{p}$

2005 Iran MO (3rd Round), 3

Tags: number theory , greatest common divisor , logarithms , geometry , number theory proposed

For each $m\in \mathbb N$ we define $rad\ (m)=\prod p_i$, where $m=\prod p_i^{\alpha_i}$. [b]abc Conjecture[/b] Suppose $\epsilon >0$ is an arbitrary number, then there exist $K$ depinding on $\epsilon$ that for each 3 numbers $a,b,c\in\mathbb Z$ that $gcd (a,b)=1$ and $a+b=c$ then: \[ max\{|a|,|b|,|c|\}\leq K(rad\ (abc))^{1+\epsilon} \] Now prove each of the following statements by using the $abc$ conjecture : a) Fermat's last theorem for $n>N$ where $N$ is some natural number. b) We call $n=\prod p_i^{\alpha_i}$ strong if and only $\alpha_i\geq 2$. c) Prove that there are finitely many $n$ such that $n,\ n+1,\ n+2$ are strong. d) Prove that there are finitely many rational numbers $\frac pq$ such that: \[ \Big| \sqrt[3]{2}-\frac pq \Big|<\frac{2^ {1384}}{q^3} \]

2006 Kyiv Mathematical Festival, 5

Tags: number theory proposed , number theory

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$

2011 Paraguay Mathematical Olympiad, 3

Tags: number theory proposed , number theory

If number $\overline{aaaa}$ is divided by $\overline{bb}$, the quotient is a number between $140$ and $160$ inclusively, and the remainder is equal to $\overline{(a-b)(a-b)}$. Find all pairs of positive integers $(a,b)$ that satisfy this.